GATE CE
$$\int\limits_C {{{{z^3} - 6} \over {3z - i}}} dz$$ is where C is |z| = 1
$$1.$$ The modular ratio considered in the working stress method depends on the type of steel used.
$$2.$$ There is an upper limit on the nominal shear stress in beams (even with shear reinforcement) due to the possibility of crushing of concrete in diagonal compression.
$$3.$$ A rectangular slab whose length is equal to its width may not be a two-way slab for some support conditions.
The TRUE statements are
At the limiting state of collapse in flexure, the force acting on the compression zone of the section is
$$1.\,\,\,\,\,\,$$ The width-to-thickness ratio limitations on the plate elements under
$$\,\,\,\,\,\,\,\,\,\,\,$$ compression in steel members are imposed by $$IS:$$ $$800$$ - $$1984$$ in order to avoid
$$\,\,\,\,\,\,\,\,\,\,\,$$ fabrication difficulties.
$$2.\,\,\,\,\,\,$$ In a doubly reinforced concrete beam, the strain in compressive reinforcement is
$$\,\,\,\,\,\,\,\,\,\,\,$$ higher than the strain in the adjoining concrete.
$$3.\,\,\,\,\,\,$$ If a cantilever $${\rm I}$$-section supports slab construction all along its length with
$$\,\,\,\,\,\,\,\,\,\,\,$$ sufficient friction between them, the permissible bending stress in compression
$$\,\,\,\,\,\,\,\,\,\,\,$$ will be the same as that in tension.
The TRUE statements are
The depth of neutral axis for the balanced failure is
$${\rm I}.\,\,\,\,$$ Effective length of a battened column is usually increased to account for the additional load on battens due to the lateral expansion of columns.
$${\rm II}.\,\,\,\,$$ As per $$IS:$$ $$800$$-$$1984,$$ permissible stress in bending compression depends on both Euler buckling stress and the yield stress of steel.
$${\rm III}.\,\,\,\,$$ As per $$IS:$$ $$800$$-$$1984,$$ the effective length of a column effectively held in position at both ends but not restrained against rotation, is taken to be greater than that in the ideal end conditions.
The TRUE statements are
$${\rm I}.\,\,\,\,\,\,\,\,\,$$ The entire cross-sectional area of the connected leg is assumed to contribute to the effective area in case of angles.
$${\rm II}.\,\,\,\,\,\,\,\,\,$$ Two angles back-to-back and tack-welded as per code requirements may be assumed to be behave as a $$T$$-section
$${\rm III}.\,\,\,\,\,\,\,\,\,$$ A check on slenderness ratio may be necessary in some cases.
The TRUE statements are

Where $$\alpha $$ is equal to

The reaction at $$C$$ is

The rotation at $$B$$ is





The beam is possibly under the action of following loads



